>>> "J. Barkley Rosser, Jr." <[EMAIL PROTECTED]> 05/25/00 11:27AM >>>
     For those who are curious, I have a recently published
paper on these issues.
"Aspects of dialectics and non-linear dynamics," _Cambridge
Journal of Economics_, May 2000, vol. 24, no. 3, pp. 311-324.
     It is also available on my website without the figures at
http://cob.jmu.edu/rosserjb.

)))))))))))

CB: In re mathematics and dialectics, Engels says in _Anti-Duhring_:

We have already noted that one of the basic principles of higher mathematics is the 
contradiction that in certain circumstances straight lines and curves may be the same. 
It also gets up this other contradiction: that lines which intersect each other before 
our eyes nevertheless, only five or six centimetres from their point of intersection, 
can be shown to be parallel, that is, that they will never meet even if extended to 
infinity. And yet, working with these and with even far greater contradictions, it 
attains results which are not only correct but also quite unattainable for lower 
mathematics.

But even lower mathematics teems with contradictions. It is for example a 
contradiction that a root of A should be a power of A, and yet A 1/2 = [--Missing 
Picture--]. It is a contradiction that a negative quantity should be the square of 
anything, for every negative quantity multiplied by itself gives a positive square. 
The square root of minus one is therefore not only a contradiction, but even an absurd 
contradiction, a real absurdity. And yet  is in many cases a necessary result of 
correct mathematical operations. Furthermore, where would mathematics -- lower or 
higher -- be, if it were prohibited from operation with ?

In its operations with variable quantities mathematics itself enters the field of 
dialectics, and it is significant that it was a dialectical philosopher, Descartes, 
who introduced this advance. The relation between the mathematics of variable and the 
mathematics of constant quantities is in general the same as the relation of 
dialectical to metaphysical thought. But this does not prevent the great mass of 
mathematicians from recognising dialectics only in the sphere of mathematics, and a 
good many of them from continuing to work in the old, limited, metaphysical way with 
methods that were obtained dialectically.

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